Fast sort engine

ABSTRACT

A method of sorting an array of data elements, the method includes assigning values associated with a monotonic function to the data elements, and sorting the array of data elements by sorting the monotonic function values using a least significant digit (LSD) Radix sort.

RELATED APPLICATION

This application claims the benefit of priority from U.S. ProvisionalPatent Application No. 62/837,780 filed 24 Apr. 2019, the contents ofwhich are incorporated herein by reference in their entirety.

FIELD AND BACKGROUND OF THE INVENTION

The present invention, in some embodiments thereof, relates to sortengines and, more particularly, but not exclusively, to a hardwareimplemented linear monotonic sort engine.

Radix sort is a non-comparative integer sorting algorithm that sortsdata with integer keys by grouping keys according to individual digitswhich share the same significant position and value. A positionalnotation is required, but because integers may be used to representstrings of characters (e.g., names or dates) and specially formattedfloating point numbers, radix sort is not limited to integers. The sortmay be implemented to start at either the most significant digit (MSD)or least significant digit (LSD). For example, when processing thenumber 1234 while sorting an array of numbers, one may start with 1 asthe MSD or with 4 as the LSD.

LSD radix sorts typically use the following sorting order: short keyscome before longer keys, and then keys of the same length are sortedlexicographically. This coincides with the normal order of integerrepresentations, such as the sequence 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11.

MSD radix sorts use lexicographic order, which is suitable for sortingstrings, such as words, or fixed-length integer representations. Asequence such as “b, c, d, e, f, g, h, i, j, ba” would belexicographically sorted as “b, ba, c, d, e, f, g, h, i, j”. Iflexicographic ordering is used to sort variable-length integerrepresentations, then the representations of the numbers from 1 to 10would be output as 1, 10, 2, 3, 4, 5, 6, 7, 8, 9, as if the shorter keyswere left-justified and padded on the right with blank characters tomake the shorter keys as long as the longest key for the purpose ofdetermining sorted order.

The radix sort may be performed using bucket sorting which is a sortingalgorithm which distributes the elements of an array into a number ofbuckets. Each bucket is then sorted individually. The buckets sortgenerally involves the following steps: (a) set up an array of initiallyempty buckets; (b) go over the original array, putting each element inits bucket; (c) sort each non-empty bucket; and (d) visit the buckets inorder and put all the elements back into the original array.

SUMMARY OF THE INVENTION

There is provided, in accordance with an embodiment of the presentinvention, a method of sorting an array of data elements, the methodincluding assigning values associated with a monotonic function to thedata elements, and sorting the array of data elements by sorting themonotonic function values using a least significant digit (LSD) Radixsort. The monotonic function may be a non-decreasing monotonic function.Alternatively, the monotonic function may be a non-increasing monotonicfunction. Optionally, the monotonic function values may be integers.

In some embodiments, the method may include creating an array of themonotonic function values. The method may additionally include creatingan array of index values, each index value corresponding to a dataelement and to its assigned monotonic function value.

In some embodiments, the method may include sorting the data elements bysorting the array of index values using the LSD Radix sort.

In some embodiments, the method may include using LSD Radix sortbuckets. The data elements may be sorted into the LSD Radix sort bucketsby sorting the assigned monotonic function values. Optionally, theassigned monotonic function values may be arranged in an array.

There is provided, in accordance with an embodiment of the presentinvention, a system for sorting an array of data elements including aprocessor, and a non-transitory computer readable medium storinginstructions executable in the processor and causing the processor toperform operations which may include assigning values associated with amonotonic function to the data elements, and sorting the array of dataelements by sorting the monotonic function values using a leastsignificant digit (LSD) Radix sort. The monotonic function may be anon-decreasing monotonic function. Alternatively, the monotonic functionmay be a non-increasing monotonic function. Optionally, the monotonicfunction values may be integers.

In some embodiments, the processor may create an array of the monotonicfunction values. The processor may also create an array of index values,each index value corresponding to a data element and to its assignedmonotonic function value. The processor may sort the data elements bysorting the array of index values using the LSD Radix sort.

In some embodiments, the processor may use LSD Radix sort buckets andmay sort the data elements into the LSD Radix sort buckets by sortingthe assigned monotonic function values. Optionally, the assignedmonotonic function values may be arranged in an array.

There is provided, in accordance with an embodiment of the presentinvention, a non-transitory computer readable medium storinginstructions executable in a processor and causing the processor toperform operations which may include assigning values associated with amonotonic function to an array of data elements, and sorting the arrayof data elements by sorting the monotonic function values using a leastsignificant digit (LSD) Radix sort.

BRIEF DESCRIPTION OF THE DRAWINGS

Some embodiments of the invention are herein described, by way ofexample only, with reference to the accompanying drawings. Details shownare for exemplary purposes and serve to provide a discussion ofembodiments of the invention. The description and the drawings may beapparent to those skilled in the art how embodiments of the inventionmay be practiced.

FIG. 1 schematically illustrates a block diagram of an exemplarycomputer system which may be used to practice the teachings disclosedherein, according to an embodiment of the present invention;

FIG. 2 schematically illustrates a block diagram of a fast sort engine(FSE), according to an embodiment of the present invention;

FIG. 3 schematically illustrates a flow chart of a method ofmonotonically sorting an array of elements using an element valueextractor module, an index sorting & generating module, and an elementvalue assigner module, all in the FSE, according to an embodiment of thepresent invention;

FIG. 4A shows an exemplary table including an IDX array with indexvalues and a VAL array with numerical values below the correspondingindex values, according to an embodiment of the present invention;

FIG. 4B which shows an exemplary table including a rearranged IDX arraywith index values and a sorted VAL array with numerical values below thecorresponding index values, according to an embodiment of the presentinvention;

FIG. 4C shows an exemplary table with the IDX array of FIG. 4A, therearranged OIDX array, and the VAL array with the numerical values inascending order, each below its corresponding OIDX, according to anembodiment of the present invention;

FIG. 5A shows an exemplary table including the IDX array and the OIDXarray transformed to show the reversal of the roles between the IDXarray and OIDX array to generate a new indices (NIDX) array, accordingto an embodiment of the present invention;

FIG. 5B shows the original IDX array, the VAL array corresponding to theIDX array, and the NIDX array with the index values which are to beassigned the corresponding numerical values in the VAL array, accordingto an embodiment of the present invention;

FIGS. 6A-6C show an example of the complete sequence of indexassignments carried out by element value assigner module, according toan embodiment of the present invention;

FIG. 7 illustrates a flow chart of a method of monotonically sortingusing a non-decreasing function an array of elements including negativenumber values using the element value extractor module, the indexsorting & generating module, the element value assigner module, and anindex shifting module, all in the FSE, according to an embodiment of thepresent invention;

FIGS. 8A and 8B show examples of the execution of the method of FIG. 7by the FSE, according to an embodiment of the present invention;

FIG. 9 is a flow chart of an exemplary method of monotonically sortingthe array of elements using an out-of-place insertion method, accordingto an embodiment of the present invention; and

FIGS. 10A and 10B schematically illustrate an exemplary operation of thefast sort engine performing a LSD radix sort directly on an input(elements) array, according to an embodiment of the present invention.

DETAILED DESCRIPTION

Before explaining at least one embodiment of the invention in detail, itis to be understood that the invention is not necessarily limited in itsapplication to the details of construction and the arrangement of thecomponents and/or methods set forth in the following description and/orillustrated in the drawings and/or the Examples. The invention iscapable of other embodiments or of being practiced or carried out invarious ways.

A function ƒ is called a monotonically non-decreasing function if forall x and y such that x≤y one has ƒ(x)≤ƒ(y), so ƒ preserves the order.Likewise, a function is call monotonically non-increasing if, wheneverx≤y one has ƒ(x)≥ƒ(y), so ƒ reverses the order.

Applicant has realized that the Radix sort, which has typically beenlimited for sorting integers and strings and therefore with limitedapplication, may be used with non-decreasing and non-increasingmonotonic functions to perform rapid sorting applicable to moderncomputational needs. Consequently, Applicant has devised a fast sortengine which applies a monotonic function on elements of an input arrayand then uses LSD radix to sort the monotonic function values andcorrespondingly the elements as well as their indices. By giving anumerical value to each element in the input array, the fast sort enginereduces the general sorting problem to a numerical sorting problem whichmay be solved with the LSD radix sort in linear runtime complexity.Since the function ƒ is monotonic, sorting the values of ƒ is equivalentto sorting the elements in the input array since the permutationsapplied to the monotonic function values array are exactly thepermutations which may be applied to the input array in order to sortit. For convenience hereinafter, “input array” may also be referred toas “elements array”.

Applicant has further realized that in lieu of using a numerical valuearray and an indices array, the fast sort engine may perform a LSD radixsort directly on the elements array. The LSD radix sort may use bucketsthat may contain elements instead of integers and may use the monotonicvalue corresponding to each element in the elements array to determineto which bucket the element will be assigned. The sort engine may sortthe elements array as it sorts the monotonic function values array.Alternatively, the sort engine may associate the monotonic value withits corresponding element and sort the elements array only, using themonotonic value of each element to determine to which bucket of the LSDradix sort each element may be assigned.

Reference is now made to FIG. 1 which schematically illustrates a blockdiagram of an exemplary computer system 100 which may be used topractice the teachings disclosed herein, according to an embodiment ofthe present invention. Computer system 100 may include a Fast SortEngine (FSE) 102, a processor 104, a cache/buffer 106, a memory 108, anetwork interface 110, an I/O interface 112, and at least one I/O device114.

FSE 102 may be used to perform rapid sorting of elements in an elementsarray by applying a monotonic function to the elements of the array andsorting both the corresponding monotonic function values and theindices. The components of FSE 102 and its functioning is described ingreater detail hereinafter with reference to FSE 200 shown in FIG. 2 andassociated description.

Processor 104 may be a computing device for executing hardwareinstructions or software, and may include those stored in memory 108.Processor 104 may be any custom made or commercially availableprocessor, a central processing unit (CPU), an auxiliary processor amongseveral processors associated with computer system 100, a semiconductorbased microprocessor (in the form of a microchip or chip set), amacroprocessor, or generally any device for executing instructions.Processor 104 may include a cache/buffer 106. Processor 104 may beconfigured to execute instructions stored within memory 108, tocommunicate data to and from the memory 108, and to generally controloperations of computer system 100 pursuant to the instructions.

Memory 108 may include any one or combination of volatile memoryelements (e.g., random access memory RAM, such as DRAM, SRAM, SDRAM,etc.) and nonvolatile memory elements (e.g., ROM, erasable programmableread only memory EPROM, electronically erasable programmable read onlymemory EEPROM, programmable read only memory PROM, tape, compact discread only memory CD-ROM, disk, diskette, cartridge, cassette or thelike, etc.). Moreover, memory 108 may incorporate electronic, magnetic,optical, and/or other types of storage media. Optionally, memory 108 mayhave a distributed architecture, where various components are situatedremote from one another, but may be accessed by processor 104.

The instructions in memory 108 may include one or more separateprograms, each of which may include an ordered listing of executableinstructions for implementing logical functions. In the example of FIG.1, the instructions in memory 108 may include any suitable operatingsystem. The operating system may essentially control the execution ofother computer programs and may provide scheduling, input-outputcontrol, file and data management, memory management, and communicationcontrol and related services.

Network interface 110 may serve to connect computer system 100 to anetwork 116. Network 116 may be an IP-based network for communicationbetween the computer system 100 and any external server, client and thelike via a broadband connection. Network 116 may transmit and receivedata between computer system 100 and external systems. Optionally,network 116 may be a managed IP network administered by a serviceprovider. Network 116 may be implemented in a wireless fashion, e.g.,using wireless protocols and technologies, such as Wi-Fi, WiMAX, etc.Network 116 may also be a packet-switched network such as a local areanetwork, wide area network, metropolitan area network, Internet network,or other similar type of network environment. Network 116 may be a fixedwireless network, a wireless local area network (LAN), a wireless widearea network (WAN) a personal area network (PAN), a virtual privatenetwork (VPN), intranet or other suitable network system and may includeequipment for receiving and transmitting signals.

I/O interface 112 may serve to output processed data to an output deviceconnected to the computer system and to receive data entry from an inputdevice, both devices shown generically in the figure as I/O device 114.I/O device 114 may include a display, a conventional keyboard and mouse,a scanner, a printer, an imaging device, a microphone, among many otherdevices which may serve to either output processed data or may be usedfor data entry. I/O device 114 may further include devices thatcommunicate both inputs and outputs, for example, a network interfacecard (NIC) or a modulator/demodulator, a radio frequency (RF) or othertransceiver, a telephonic interface, a bridge, a router, and the like.

Reference is now made to FIG. 2 which schematically illustrates a blockdiagram of FSE 200, according to an embodiment of the present invention.FSE 200 may include a processor 202, a memory 204, a cache/buffer 206,an element value extractor module 208, an index sorting & generatingmodule 210, an element value assigner module 212, and an optional indexshifting module 214. FSE 200 and FSE 102 in FIG. 1 may include the samecomponents and may perform the same functions.

The operation of FSE 200 may be described in greater detail withreference to FIGS. 3-10B. A non-decreasing monotonic function may beapplied on the elements of an array and then LSD radix may be used tosort the monotonic function values and optionally order the indicesassociated with the elements in the array. The monotonic function may beselected so that f (x) returns integer numbers. It may be readilyappreciated by the skilled person that, although the operation isdescribed with reference to use of a non-decreasing monotonic function,a non-increasing monotonic function may also be used in lieu of thenon-decreasing monotonic function.

In some embodiments, a function g(x) which returns floating point valuesmay be required. In these cases, for example, the function g(x) may beconverted to a function that returns integer values and may remainmonotonic by returning the integer value which corresponds to thefloating—point value binary representation. If the floating-point valueis negative, the function may remain monotonic by returning the oppositenumber of the integer value which corresponds to the binaryrepresentation of the opposite number of the floating-point value (thevalues may be different).

In some embodiments, a method of the present invention may include useof two separate arrays. A first array may hold index values which maypoint to a second array which may hold monotonic function numericalvalues corresponding to the elements, as described further on below withreference to FIGS. 3-9. The first array may be referred to hereinafteras indices array and the second array as numerical value array.Alternatively, as described with reference to FIGS. 10A and 10B furtheron below, instead of the indices array, the first array may be the inputarray itself which holds the elements and the second array may be thenumerical value array. For convenience hereinafter, “monotonic functionnumerical value” may be used interchangeably with “monotonic functionvalue” and “numerical value”.

Processor 202 may control the operation of all components in the FSEincluding data flow between memory 204, cache/buffer 206, and themultiple modules 208-214. Processor 202 may additionally control all FSE200 component operations as required to sort the array of elementsstored in memory 204. Processor 202 may additionally interface withprocessor 104 in computer system 100 for data transfer between the FSEand other components of the computer system. In some embodiments, thefunctions carried out by processor 202 may be provided by processor 104.

Memory 204 may store an unsorted input array of unsorted elements priorto, and during the monotonic sorting operation. It may additionallystore the sorted array following monotonic sorting. Memory 204 mayadditionally include executable instructions associated with theoperation of FSE 200. Optionally, the functions carried out by memory204 may be provided by memory 108. Cache/buffer 206 may temporarilystore the monotonic function value associated with an element during thesorting operation. Optionally, the functions carried out by cache/buffer206 may be provided by cache/buffer 106 in computer system 100.

The actual monotonic sorting operation is carried out by element valueextractor module 208, sorting & generating module 210, element valueassigner module 212, and optional shifting module 214. Reference is nowalso made to FIG. 3 which schematically illustrates a flow chart of amethod 300 of monotonically sorting an array of elements with modules208-212 using an index array and a monotonic numerical value array,according to an embodiment of the present invention. Use of shiftingmodule 214 together with modules 208-212 will be described later on withreference to FIG. 7.

At 302, element value extractor module 208 may apply the monotonicfunction to the elements, may build the numerical value array, and mayextract the monotonic function numerical value (VAL) associated witheach of the unsorted elements from the numerical value array accordingto the indices (IDX) array. The extraction may be sequential and mayfollow the order of the indices in the IDX array (e.g. ascending order).An example of this operation is shown in FIG. 4A which shows anexemplary table 400 including the IDX array 402 with the index valuesand the VAL array 404 with the monotonic function numerical valueassociated with each of the elements below the corresponding indexvalue.

At 304, sorting and generating module 210 may sort the numerical valuesin the numerical value array in numerical order (e.g. ascending order)according to the VAL. It may correspondingly rearrange the IDX in theindices array accordingly to generate an “ordered” indices (OIDX) array.Each permutation made on the numerical value array may correspondinglybe made on the elements array and on the indices array as well. Anexample of the rearranging operation is shown in FIG. 4B which shows anexemplary table 410 including the rearranged IDX array 412 with theindex values and the sorted VAL array 414 with the monotonic functionnumerical value below the corresponding index value. VAL array 414 isarranged in numerically ascending order. FIG. 4C shows an exemplarytable 420 with the original IDX array 402, the rearranged OIDX array412, and the VAL array 414 with the monotonic function numerical valuesin ascending order, each below its corresponding OIDX.

At 306, sorting and generating module 210 may transform IDX and OIDX byreversing their roles to generate a new indices (NIDX) array. An exampleof the transformation operation is shown in FIG. 5A which shows anexemplary table 500 including IDX array 402 and OIDX array 412transformed into table 510 which shows the reversal of the roles betweenthe IDX array 402 and OIDX 412 to generate a new indices (NIDX) array512. For example, IDX=3, OIDX=0, indicated by 502 is transformed toIDX=0, NIDX=3, indicated by 514.

At 308, element value assigner module 212 may assign the elements in theelements array and their corresponding numerical values in the numericalvalue array associated with the original IDX array the corresponding newindex value in the NIDX array. An example, of the assignment is shown inFIG. 5B which shows the original IDX array 402, the VAL array 404corresponding to the IDX array, and the NIDX array 512 with the indexvalues which are to be assigned the corresponding numerical values inthe VAL array. For example, as shown in 522, VAL=4 having an originalIDX=2 may now be assigned NIDX=0.

Reference is now also made to FIGS. 6A-6C which show an example of thecomplete sequence of index assignments carried out by element valueassigner module 212, according to an embodiment of the presentinvention. Shown in table 600 are IDX array 402, VAL array 404, and NIDX512 in an initial state as per table 520 in FIG. 5B. It is noted thatevery permutation made includes the same permutation in the elementsarray.

As previously described with reference to 308, all the numerical valuesin VAL array 404 may have their corresponding index values in IDX array402 replaced by the index values in NIDX array 512. That is, VAL=15 maybe assigned an index value of 3 instead of 0, VAL=22 may be assigned anindex value of 4 instead of 1, VAL=4 may be assigned an index value of 0instead of 2, VAL=13 may be assigned an index value of 2 instead of 3,VAL=78 may be assigned an index value of 7 instead of 4, VAL=11 may beassigned an index value of 1 instead of 5, VAL=37 may remain with itsprevious index value of 6, and VAL=36 may be assigned an index value of5 instead of 7.

Shown in table 602 is, starting with the first index value IDX=0 in IDXarray 402, the assignment of VAL=15 in VAL array 404 to IDX=3 in IDXarray 402. As the numerical value has now been assigned to IDX array 402a null (“X”) is placed in NIDX array 512. Furthermore, as IDX=3 in IDXarray 402 was previously assigned to VAL=13 and now it corresponds toVAL=15, VAL=13 is placed in a buffer 650.

Shown in table 604 is the assignment of the value in buffer 650, VAL=13to IDX=2 in IDX array 402. As the numerical value has now been assignedto IDX array 402 a null (“X”) is placed in NIDX array 512. Furthermore,as IDX=2 in IDX array 402 was previously assigned to VAL=4 and now itcorresponds to VAL=13, VAL=4 is placed in buffer 650.

Shown in table 606 is the assignment of the value in buffer 650, VAL=4to IDX=0 in IDX array 402. As the numerical value has now been assignedto IDX array 402 a null (“X”) is placed in NIDX array 512. Furthermore,as IDX=0 in IDX array 402 was previously assigned a null (“X”) whenVAL=15 was assigned (as indicated by “X”), no VAL is placed in buffer650.

Shown in table 608 is the assignment of the value VAL=22 correspondingto the next sequential index value IDX=1 in IDX array 402 to IDX=4 inthe array. As the numerical value has now been assigned to IDX array 402a null (“X”) is placed in NIDX array 512. Furthermore, as index value=4in IDX array 402 was previously assigned to VAL=78 and now itcorresponds to VAL=22, VAL=78 is placed in buffer 650.

Shown in table 610 is the assignment of the value in buffer 650, VAL=78to IDX=7 in IDX array 402. As the numerical value has now been assignedto IDX array 402 a null (“X”) is placed in NIDX array 512. Furthermore,as IDX=7 in IDX array 402 was previously assigned to VAL=36 and now itcorresponds to VAL=78, VAL=36 is placed in buffer 650.

Shown in table 612 is the assignment of the value in buffer 650, VAL=36to IDX=5 in IDX array 402. As the numerical value has now been assignedto IDX array 402 a null (“X”) is placed in NIDX array 512. Furthermore,as IDX=5 in IDX array 402 was previously assigned to VAL=11 and now itcorresponds to VAL=36, VAL=11 is placed in buffer 650.

Shown in table 614 is the assignment of the value in buffer 650, VAL=11to IDX=1 in IDX array 402. As the numerical value has now been assignedto IDX array 402 a null (“X”) is placed in NIDX array 512. Furthermore,as IDX=1 in IDX array 402 was previously assigned a null (“X”) whenVAL=22 was assigned (as indicated by “X”), no VAL is placed in buffer650.

Shown in table 616 is the assignment of the value VAL=36 correspondingto the next sequential index value which has not been assigned, IDX=6 inIDX array 402. As may be appreciated from the table NIDX=6 in NIDX array512 which is the same as IDX=6 in IDX array 402, therefore no assignmentis required. A null (“X”) is placed in NIDX array 512 as shown in table618.

Shown in table 618 are both IDX array 402 and the VAL array 404monotonically sorted in a non-decreasing arrangement, the result of theexecution of the method of FIG. 3. It may then be appreciated that themonotonic function numerical values in the numerical value arraycorresponding to the elements in the elements array have been sortedusing the monotonic non-decreasing function. Applying the operationspresented in FIGS. 6A-6C to the elements array instead of the numericalvalues array, using the same IDX and NIDX values, may sort the elementsarray.

Applicant has further realized that the monotonic sort performed by theFSE using the method of FIG. 3 may not properly sort the numerical valuearray if negative values are used in the array. This may be due of theuse of two's complement in the binary representation of the negativenumerical values. As the most significant bit (MSB) in the negativenumerical value is MSB=1, the LSD radix sort performed in method 300 inFIG. 3 at 302 and 304 may place the negative numbers at the end of thesorted of the OIDX array.

Applicant has further realized that the above problem when sortingnegative numerical values may be solved by shifting the NIDX values inthe generated NIDX (method 300 in FIG. 3 at 306). All NIDX which maypoint to negative numerical values in the numerical value array may beshifted forward to the beginning of the array by adding a negative shiftto each one of the NIDX values: the total number of NIDX pointing tonon-negative numerical values. All NIDX which may point to non-negativenumerical values in the numerical value array may be shifted backwardsto the end of the array by adding to each one of the NIDX values thetotal number of NIDX pointing to negative numerical values. Optionally,the forward and backward shift may be determined by counting the numberof cells in the number value array with non-negative numerical valuesand negative values, respectively.

Reference is now made to FIG. 7 which illustrates a flow chart of amethod 700 of monotonically sorting, using a non-decreasing function, anarray of elements including negative number values using modules208-214, according to an embodiment of the present invention. Referenceis also made to FIGS. 8A and 8B which show examples of the execution ofmethod 700 by FSE 200, according to an embodiment of the presentinvention.

At 702, element value extractor module 208 may apply the monotonicfunction to the elements and may extract from the numerical value arraythe numerical value (VAL) associated with the unsorted elements in theelements array according to the indices (IDX) array. The extraction maybe sequential and may follow the order of the indices in the IDX array(e.g. ascending order). An example of this operation is shown in anexemplary table 800 including the IDX array 806 with the index values,the VAL array 808 with the numerical values VAL corresponding to eachIDX and including negative numerical values, and the binary array 810including the binary representation for each numerical value. As may beappreciated, in the table, the binary representation for the negativenumbers uses the two's complements method.

At 704, sorting and generating module 210 may sort the VAL in thenumerical value array in numerical order (e.g. ascending order) and maycorrespondingly rearrange the IDX in the indices array accordingly togenerate an “ordered” indices (OIDX) array. Each permutation made on thenumerical value array may be made on the indices array as well. Anexample of the rearranging operation is shown in an exemplary table 802which shows IDX array 806, OIDX array 812, sorted VAL array 808, andsorted binary representation array 810. It may be appreciated from table802 that the negative numbers have been sorted to the bottom of thetable as the LSD radix sort is affected from the binary representationand the two's complements method.

At 706, sorting and generating module 210 may transform IDX and OIDX byreversing their roles to generate a new indices (NIDX) array. An exampleof the transformation operation is shown in an exemplary table 804 whichshows the reversal of the roles between the IDX array 806 and OIDX 812in table 802 to generate a new indices (NIDX) array 814. For example,IDX=3, OIDX=4, indicated by 816 is transformed to IDX=4, NIDX=3,indicated by 818.

At 708, shifting module 214 may calculate the shift 820 to be applied toeach NIDX value in NIDX array 814. For example, as there are 3non-negative numerical values and 2 negative numerical values, the shiftis −3 for NIDX pointing to negative numerical values and +2 for NIDXpointing to non-negative numerical values in numerical value array 808,as shown in shift array 820.

At 710, shifting module 214 may generate a new shift IDX array 822including shift IDX values by adding to each NIDX value in NIDX array814 the negative or non-negative shift value in shift array 820. Thisnew shift IDX array 822 now points to the corresponding numerical valuesin numerical value array in a way that places the negative numericalvalues in the beginning of the array.

At 712, element value assigner module 212 may assign the numerical valuein the original IDX array the corresponding new index value in the shiftIDX array. An example of the assignment is shown in FIG. 8B at table 805which shows the original IDX array 806, the VAL array 808 correspondingto the IDX array, and the shift IDX array 822 with the index valueswhich are to be assigned the corresponding numerical values in the VALarray. For example, as shown in 824, VAL=−10 having an original IDX=4may now be assigned shift IDX=0. The complete sequence of indexassignments carried out by element value assigner module 212 may followa similar procedure to that shown in FIGS. 6A-6C with the exception thatthe NIDX array 512 in the figure may be replaced with the shift IDXarray 822 in FIGS. 8A and 8B.

Applicant has additionally realized that the fast sort engine may use anout-of-place insertion method to do parallel sorting of an input arrayin one or more CPUs. Similarly to the previously described monotonicallysorting method, an OIDX array is generated but instead of generating aNIDX and making in-place assignments, an auxiliary array may be createdwith the OIDX in a different area of the memory. That is, the OIDX mayserve as the NIDX in the previously described method. The method may beparticularly advantageous as it does not make in-place assignments onthe elements array. For example, if there is an array with 20 elementswhere there are 10 monotonic values that are smaller than X and 10monotonic values that are larger than X, they may be sorted in paralleland the results may be copied to the elements array. Elements in theelements array associated with monotonic values larger than X mustfollow those that are smaller than X because the monotonic functionpreserves the order. Consequently, the elements with monotonic valuesthat are smaller than X may be copied to the first 10 places in theelements array and the elements with monotonic values that are largerthan x to the next 10 places in the elements array. Alternatively theelements array may be split arbitrarily into several sub-arrays whichmay be sorted in parallel and then merged into the elements array.

Reference is now made to FIG. 9 which is a flow chart of an exemplarymethod 900 of monotonically sorting the array of elements using theout-of-place insertion method, according to an embodiment of the presentinvention. In performing the out-of-place insertion method, some or allof the components shown in the block diagram of FSE 200 may be used,optionally additional components may be used including additionalprocessors 202.

At 902, the same actions described at 302 of FIG. 3 are performed.

At 904, the same actions described at 304 of FIG. 3 are performed.

At 906, the OIDX array may be written into a different section of memory204.

At 908, rearrange the numerical values in the OIDX array into thecorresponding IDX array. Referring back to FIGS. 4A-4C, the elementassociated with monotonic function value 4 and OIDX=2 may now beassigned to IDX=0 (in the auxiliary array); the element associated withmonotonic function value 11 and OIDX=5 may now be assigned to IDX=1 (inthe auxiliary array); the element associated with monotonic functionvalue 13 and OIDX=3 may now be assigned to IDX=2 (in the auxiliaryarray); the element associated with monotonic function value 15 andOIDX=0 may now be assigned to IDX=3 (in the auxiliary array); theelement associated with monotonic function value 22 and OIDX=1 may nowbe assigned to IDX=4 (in the auxiliary array); the element associatedwith monotonic function value 36 and OIDX=7 may now be assigned to IDX=5(in the auxiliary array); the element associated with monotonic functionvalue 37 and OIDX=6 may now be assigned to IDX=6 (in the auxiliaryarray); and the element associated with monotonic function value 78 andOIDX=4 may now be assigned to IDX=7 (in the auxiliary array). Finallythe auxiliary array may be copied to the elements array.

For negative monotonic function number values, the shifting processdescribed with reference to FIGS. 8A and 8B may be similarly performedfor the out-of-place insertion method using OIDX instead of NIDX. Theshift and SHIFT IDX may be similarly computed as described withreference to the mentioned figures.

Reference is now made to FIGS. 10A and 10B which schematicallyillustrate an exemplary operation of fast sort engine 200 performing aLSD radix sort directly on an input (elements) array, according to anembodiment of the present invention. In performing the LSD radix directsort method, some or all of the components shown in the block diagram ofFSE 200 may be used, including additional components such as, forexample, one or more processors 202. Furthermore, sorting and generatingmodule 210 and shifting module 214 may perform sorting and shiftingfunctions on the input array, respectively, some of which may be similarto those previously described with reference to the index array.Additionally, although the operation of fast sort engine 200 isdescribed herein with reference to making permutations on a monotonicfunction values array, in some embodiments, the operation of the fastsort engine may not include creating the monotonic function values arrayrather associating the values with its corresponding element and onlysorting the input array.

In FIG. 10A may be seen a first step in the exemplary LSD radix directsort operation performed on an exemplary elements (ELMT) array 1004having three elements A, B, C, and a monotonic function values (VAL)array 1006 having values of 93, 43, and 12. Element A occupies row 1008in ELMT array 1004 and is assigned the monotonic function value 93,element B occupies row 1010 in the elements array and is assigned themonotonic function value 43, and element C occupies row 1012 in theelements array and is assigned the monotonic function value 12. Tenempty buckets 1014 labelled “Bucket 0” through “Bucket 9” are used toperform the LSD radix direct sort operation.

In a first sort step, as indicated by arrow 1018, the elements aresorted into the buckets according to the units digit of thecorresponding numerical value which is the LSD. The ten bucketsincluding the elements, shown as buckets 1016, now hold in Bucket 2 theelement C as its corresponding monotonic value is 12, indicated as C/121013; and in Bucket 3 the elements A and B as their correspondingmonotonic values are 93 and 43, indicated as A/93 1009 and B/43 1011,respectively. Following the first sort step, the elements are thencopied from the buckets back into the ELMT 1004 following the order ofthe buckets, as shown by arrow 1020, so that row 1008 in the elementsarray 1004 now holds element C, row 1010 holds element A, and row 1012holds element B.

In FIG. 10B may be seen a second and final step in the exemplary LSDradix direct sort operation performed on the exemplary ELMT array 1004.In this step, the elements in ELMT array 1004 from the end of theprevious step are sorted into the buckets according to the tens digit ofthe corresponding numerical value which is now the next LSD. The tenbuckets including the elements, shown as buckets 1016, now hold inBucket 1 the element C as its corresponding monotonic value is 12,indicated as C/12 1013; in Bucket 4 the element B as its correspondingmonotonic values is 43, indicated as B/43 1011; and Bucket 9 the elementC as its corresponding monotonic values is 93, indicated as A/43 1009.Following this second and final sort step, the elements are then copiedfrom the buckets back into the ELMT array 1004 following the order ofthe buckets, as shown by arrow 1020, so that row 1008 in the elementsarray 1004 now holds element C, row 1010 holds element B, and row 1012holds element C, and the input array has been sorted. For negativemonotonic function number values, the elements corresponding to thenegative monotonic values may be copied to a temporary array in the sameorder they reside in the elements array, and the elements correspondingto the non-negative monotonic values may be shifted towards the end ofthe elements array. The elements corresponding to the negative monotonicvalues may then be copied from the temporary array to the beginning ofthe elements array in the same order they reside in the temporary array.Optionally, the size of the shift may be determined by counting thenumber of elements in the elements array corresponding to negativemonotonic values. Alternatively, the elements corresponding to thenon-negative values may be copied to the temporary buffer and theelements that correspond to the negative monotonic values may be shiftedto the beginning of the array. For example, if there is an array with 20elements where there are 5 elements corresponding to negative monotonicvalues, after performing the LSD radix sort on the array, the 5 elementscorresponding to the negative monotonic values may be copied to atemporary array and the remaining 15 elements may be pushed 5 placestowards the end of the array. The elements in the temporary array maythen be copied to the beginning of the array and occupy the 5 firstplaces.

The fast sort engine operation previously described in FIGS. 10A and 10Bused ten buckets for exemplary purposes. The skilled person may readilyappreciate that the fast sort engine operation may include use of agreater number of buckets, for example 256 buckets which may correspondwith the number of bits in a byte. For words with lengths greater than abyte, for example, a 16-bit word, a 32-bit word, or a 64-bit word, thewords may be split into bytes and the LSD radix sort may be performed oneach byte, optionally on a group of bytes.

Unless specifically stated otherwise, as apparent from the precedingdiscussions, it is appreciated that, throughout the specification,discussions utilizing terms such as “processing,” “computing,”“calculating,” “determining,” or the like, refer to the action and/orprocesses of a computer, computing system, or similar electroniccomputing device that manipulates and/or transforms data represented asphysical, such as electronic, quantities within the computing system'sregisters and/or memories into other data similarly represented asphysical quantities within the computing system's memories, registers orother such information storage, transmission or display devices.

Embodiments of the present invention may include apparatus forperforming the operations herein. This apparatus may be speciallyconstructed for the desired purposes, or it may comprise ageneral-purpose computer selectively activated or reconfigured by acomputer program stored in the computer. Such a computer program may bestored in a computer readable storage medium, such as, but not limitedto, any type of disk, including floppy disks, optical disks,magnetic-optical disks, read-only memories (ROMs), compact discread-only memories (CD-ROMs), random access memories (RAMs),electrically programmable read-only memories (EPROMs), electricallyerasable and programmable read only memories (EEPROMs), magnetic oroptical cards, Flash memory, or any other type of media suitable forstoring electronic instructions and capable of being coupled to acomputer system bus.

The processes and displays presented herein are not inherently relatedto any particular computer or other apparatus. Various general-purposesystems may be used with programs in accordance with the teachingsherein, or it may prove convenient to construct a more specializedapparatus to perform the desired method. The desired structure for avariety of these systems will appear from the description below. Inaddition, embodiments of the present invention are not described withreference to any particular programming language. It will be appreciatedthat a variety of programming languages may be used to implement theteachings of the invention as described herein.

The foregoing description and illustrations of the embodiments of theinvention has been presented for the purposes of illustration. It is notintended to be exhaustive or to limit the invention to the abovedescription in any form.

Any term that has been defined above and used in the claims, should beinterpreted according to this definition.

1. A method of sorting an array of data elements, the method comprising:assigning values associated with a monotonic function to the dataelements; and sorting the array of data elements by sorting saidmonotonic function values using a least significant digit (LSD) Radixsort.
 2. A method according to claim 1 wherein said monotonic functionis a non-decreasing monotonic function.
 3. A method according to claim 1wherein said monotonic function is a non-increasing monotonic function.4. A method according to claim 1 further comprising creating an array ofsaid monotonic function values.
 5. A method according to claim 1 furthercomprising creating an array of index values, each index valuecorresponding to a data element and to its assigned monotonic functionvalue.
 6. A method according to claim 5 comprising sorting the dataelements by sorting said array of index values using the LSD Radix sort.7. A method according to claim 1 further comprising using LSD Radix sortbuckets.
 8. A method according to claim 7 comprising sorting the dataelements into said LSD Radix sort buckets by sorting said assignedmonotonic function values.
 9. A method according to claim 7 wherein saidassigned monotonic function values are arranged in an array.
 10. Amethod according to claim 1 wherein said assigned monotonic functionvalues are integers.
 11. A system for sorting an array of data elementscomprising: a processor; and a non-transitory computer readable mediumstoring instructions executable in said processor and causing saidprocessor to perform operations comprising: assigning values associatedwith a monotonic function to the data elements; and sorting the array ofdata elements by sorting said monotonic function values using a leastsignificant digit (LSD) Radix sort.
 12. A system according to claim 11wherein said monotonic function is a non-decreasing monotonic function.13. A system according to claim 11 wherein said monotonic function is anon-increasing monotonic function.
 14. A system according to claim 11further comprising said processor creating an array of said monotonicfunction values.
 15. A system according to claim 11 further comprisingsaid processor creating an array of index values, each index valuecorresponding to a data element and to its assigned monotonic functionvalue.
 16. A system according to claim 15 comprising said processorsorting the data elements by sorting said array of index values usingthe LSD Radix sort.
 17. A system according to claim 11 furthercomprising said processor using LSD Radix sort buckets.
 18. A systemaccording to claim 17 comprising said processor sorting the dataelements into said LSD Radix sort buckets by sorting said assignedmonotonic function values.
 19. A system according to claim 17 whereinsaid assigned monotonic function values are arranged in an array.
 20. Anon-transitory computer readable medium storing instructions executablein a processor and causing the processor to perform operationscomprising: assigning values associated with a monotonic function to anarray of data elements; and sorting said array of data elements bysorting said monotonic function values using a least significant digit(LSD) Radix sort.